I have a few queries regarding the normalizer/centralizer theorem and I would prefer a response suitable for a beginner.
Suppose that $G$ is a group and $S \subseteq G$ then we know that both $N_{G}(S)$ and $C_{G}(S)$ are subgroups of $G$ and in particular $C_{G}(S)$ is a normal subgroup of $N_{G}(S)$.
The N/C theorem only deals with the case when the subset $S$ is actually a subgroup of $G$ and not just any subset of $G$. The quotient $N(S)/G(S)$ makes sense for arbitrary subsets $S$ of $G$.
Why does the $N/C$ theorem not deal with such subsets $S$? One obvious reason is that we can't talk about $Aut(S)$ unless $S$ itself is a group. But apart from that is there any other specific reason to choose $S$ as a subgroup of $G$? Or perhaps the quotient $N/C$ is interesting enough (i.e. it provides valuable information about $S$) only when $S$ is a subgroup.
Next consider the usual N/C theorem: $$\frac{N_{G}(H)}{C_{G}(H)} \cong K \leq Aut(H)$$ where $H\leq G$. Now the members of $K$ are inner automorphisms of $G$ based on conjugation by members of $N_{G}(H)$ i.e. $K \subseteq Inn(G)$. The only reason this $K$ is also a subset of $Aut(H)$ is because members of $K$ are conjugation via members of $N_{G}(H)$.
Is my understanding about $K$ above correct? Also does $K \leq Inn(G)$?
Best Answer
Edit (thanks to Mikko and Derek) For any set $S \subseteq G$, we have $C_G(S)=C_G(\langle S \rangle)$. For the normalizer $N_G(S)\subseteq N_G(\langle S \rangle)$ and this inclusion can be strict. Hence $N_G(S)/C_G(S)$ is a subgroup $N_G(\langle S \rangle)/C_G(S)$ but the first embeds in $S_{|S|}$. The latter in $Aut(\langle S \rangle)$, which in general is a much smaller group, which makes it more suitable for analysis of the structure of $G$.